2.2 Slope and Rate of Change

2.2 Slope and Rate of Change

What is Slope?•Slope is the slant of a line.•Represented by the letter m•The slope of a non-vertical line passing through points (x1, y1) and (x2, y2) is:

= rise run

• http://www.youtube.com/watch?v=IyfpM-ruafo

Example:

•Find the slope of the line passing through the points (-1,-1) and (1,2).

Your Turn!

•Find the slope of the line passing through the points (-3, 5) and (2, 1).

Classifying Lines by Slope

positive

m > 0

zerom = 0

negativem < 0

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Classifying Lines Without Graphing

•Calculate the slope, then decide if the line rises, falls, is horizontal, or is vertical.

•Example: The line through ( 3, -4) and (1, -6).

•Example: The line through ( 2, -1) and (2, 5).

Comparing the Steepness of Lines•For two lines with positive slopes, the line with the larger slope is steeper.

•For two lines with negative slopes, the line with the smaller slope (the more negative) is steeper.

activity

Example:

•Tell which line is steeper.Line 1: through (2,3) and (4,7)Line 2: through (-1,2) and (4,5)

Your Turn!

•Tell which line is steeper.Line 1: through (-1,-3) and (-3,-2)Line 2: through (3,-4) and (0,-3)

What are Parallel and Perpendicular Lines?

•Parallel lines lie in the same plane and never intersect.

•Perpendicular lines lie in the same plane and intersect at a right angle.

•We can use slope to determine whether two lines are parallel or perpendicular.

Parallel Lines•Two non-vertical lines are parallel if and only if they have the same slope.

m1 = m2

Perpendicular Lines•Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other.

m1 = - 1 or m1m2 = -1

m2

Example:•Tell whether the lines are parallel, perpendicular, or neither.

Line 1: through (-3,3) and (3,-1)Line 2: through (-2,-3) and (2,3)

Your Turn!

•Tell whether the lines are parallel, perpendicular, or neither.

Line 1: through (-2,-2) and (4,1)Line 2: through (-3,-3) and (1,5)

stop

Using Slope In Real Life•Example 1: For safety reasons, the distance from the base of a ladder to the wall should be at least one quarter of the height where the ladder’s top hits the wall. For example, a ladder that hits the wall at 12 feet should have its base out 3 feet from the wall.

a. Find the maximum recommended slope for a ladder.

b. Find the maximum distance a ladder’s base should be from a wall if you need the ladder to reach a height of 20 feet.

•Example 2:•The slope, or “grade”, of a road is expressed as a percent. For example, if a road has a 3% grade, it rises 3 feet for every 100 feet of horizontal distance.

a.Find the grade of a road that rises 75 feet over a horizontal distance of 2000 feet.

b.Find the horizontal length x of a road with a grade of 4% if the road rises 50 feet over its length.

Your Turn!•A water park slide drops 8 feet over a horizontal distance of 24 feet. Find its slope. Then find the drop over a 54 foot section with the same slope.

Slope as a Rate of Change

•In real-life problems we often use slope to describe and average rate of change.

•These rates use units like “miles per hour” or “dollars per year”

•What is another unit used for a rate of change?

Example:•In the Mojave Desert in California, temperatures can drop quickly from day to night. Suppose the temperature drops from 100 degrees at 2 pm to 68 degrees at 5 am. Find the average rate of change and use it to determine the temperature at 10 pm.

Example 2:•The number of U.S. cell phone subscribers increased from 16 million in 1993 to 44 million in 1996. Find the average rate of change and use it to estimate the number of subscribers in 1997.

Your Turn!

•The average monthly cell phone bill decreased from $61.48 in 1993 to $47.70 in 1996. Find the average rate of change and use it to estimate the average monthly bill in 1997.

Your Turn!•You are driving on a road trip through Europe. If at 9 am you are 420 km from Rome and at 3 pm you are 108 km from Rome, what is your average speed?